Start with circle with center C and chord AB where A and B are random points on a circle. Draw radii, R, from C to A and from C to B. Designate angle ACB as alpha. Bisect angle alpha with a line from C to D, the midpoint of AB. Therefore, Sin(alpha/2) = AD/AC = AD/R. Or, Rsin(alpha/2) = AD. Since AD = AB/2, we now have 2Rsin(alpha/2)= AB. So the question is how often does 2Rsin(alpah/2)exceed R. Space is short so I will continue on the following sheet.